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SPEPS Preprint

Author(s): Anureet Saxena, Carnegie Mellon University
Title: A Short Note on the Probabilistic Set Covering Problem
Date of Acceptance: 29.05.2007
Submission Date: 01.03.2007
Series Title: Stochastic Programming E-Print Series
(SPEPS)
Editors: Julie L. Higle; Werner Römisch; Surrajeet Sen
Complete Preprint: pdf (urn:nbn:de:kobv:11-10078066)
Keywords (eng): Probabilistic Programming, Set Covering, Mixed Integer Programming, Cutting Planes
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Abstract (eng):
In this paper we address the following probabilistic version (PSC) of the set covering problem: $ min{cx | P(Ax ≥ ξ) ≥ p, x_j \in {0, 1} j \in N }$ where A is a 0-1 matrix, ξ is a random 0-1 vector and $p \in (0, 1]$ is the threshold probability level. In a recent development Saxena, Goyal and Lejeune proposed a MIP reformulation of (PSC) and reported extensive computational results with small and medium sized (PSC) instances. Their reformulation, however, suffers from the curse of exponentiality − the number of constraints in their model can grow exponentially rendering the MIP reformulation intractable for all practical purposes. In this paper, we give a polynomial-time algorithm to separate the (possibly exponential sized) constraint set of their MIP reformulation. Our separation routine is independent of the specific nature (concave, convex, linear, non-linear etc) of the distribution function of ξ, and can be easily embedded within a branch-and-cut framework yielding a distribution-free algorithm to solve (PSC). The resulting algorithm can solve (PSC) instances of arbitrarily large block sizes by generating only a small subset of constraints in the MIP reformulation and verifying the remaining constraints implicitly. Furthermore, the constraints generated by the separation routine are independent of the coefficient matrix A and cost-vector c thereby facilitating their application in sensitivity analysis, re-optimization and warm-starting (PSC). We give preliminary computational results to illustrate our findings on a test-bed of 40 (PSC) instances created from OR-Lib set-covering instance scp41.
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